Mathematica Plotting k^2 in Mathematica - Output Not 1?

[member 428835]

Can anyone confirm if the following in Mathematica gives an output that is not 1? I'm getting some sort of sinusoid, but I should get 1.

k = 2;
Plot[k^2 JacobiSN[t, k]^2 + JacobiDN[t, k]^2, {t, 0, 10}]

Thanks!

 

[DrClaude]

Can anyone confirm if the following in Mathematica gives an output that is not 1? I'm getting some sort of sinusoid, but I should get 1.

k = 2;
Plot[k^2 JacobiSN[t, k]^2 + JacobiDN[t, k]^2, {t, 0, 10}]

Thanks!

Yes, that oscillates between 1 and 2 (Mathematica 12.0.0.0).

 

[member 428835]

Yes, that oscillates between 1 and 2 (Mathematica 12.0.0.0).

But the addition theorems here https://en.wikipedia.org/wiki/Jacobi_elliptic_functions imply $k^2 sn^2 + dn^2 = 1$? Am I missing something?

 

[Bill Simpson]

I don't know if this helps, but

Plot[2 JacobiSN[t, k]^2 + JacobiDN[t, k]^2, {t, 0, 10}]

does appear to be 1 for all values of t while 4 JacobiSN[t, k]^2 + JacobiDN[t, k]^2 is not.

I noticed that by plotting the two functions separately and guessing the needed scale factor.

 

[member 428835]

I don't know if this helps, but

Plot[2 JacobiSN[t, k]^2 + JacobiDN[t, k]^2, {t, 0, 10}]

does appear to be 1 for all values of t while 4 JacobiSN[t, k]^2 + JacobiDN[t, k]^2 is not.

I noticed that by plotting the two functions separately and guessing the needed scale factor.

Do you mean $k^2$ instead of 4?

 

[DrClaude]

The problem is that the second argument in JacobiSN and JacobiDN is $m$, not $k$ ($k = \sqrt{m}$).

See
https://www.johndcook.com/blog/2018/10/12/jacobi-function-nomenclature/